let P be Subset of (TOP-REAL 2); :: thesis: for s1 being Real st P = { |[s,r]| where s, r is Real : s <= s1 } holds

P is closed

let s1 be Real; :: thesis: ( P = { |[s,r]| where s, r is Real : s <= s1 } implies P is closed )

reconsider P1 = { |[s,r]| where s, r is Real : s > s1 } as Subset of (TOP-REAL 2) by Lm10;

assume P = { |[s,r]| where s, r is Real : s <= s1 } ; :: thesis: P is closed

then A1: P = P1 ` by Th11;

P1 is open by JORDAN1:20;

hence P is closed by A1, TOPS_1:4; :: thesis: verum

P is closed

let s1 be Real; :: thesis: ( P = { |[s,r]| where s, r is Real : s <= s1 } implies P is closed )

reconsider P1 = { |[s,r]| where s, r is Real : s > s1 } as Subset of (TOP-REAL 2) by Lm10;

assume P = { |[s,r]| where s, r is Real : s <= s1 } ; :: thesis: P is closed

then A1: P = P1 ` by Th11;

P1 is open by JORDAN1:20;

hence P is closed by A1, TOPS_1:4; :: thesis: verum